A unit, recall, is whatever we call one. (Lesson 1.) Each unit fraction is a part of number 1.
1 3 |
is the third part of 1. |
1 4 |
is the fourth part of 1. |
And so on.
Example 1. In the fraction |
4 5 |
, what number is the unit, and how many |
of them are there?
Answer. The denominator of a fraction names the unit -- the part of 1. The numerator tells their number -- how many.
In the fraction |
4 5 |
, the unit is |
1 5 |
. And there are 4 of them. |
Example 2. Let |
1 3 |
be the unit, and count to 2 |
1 3 |
. |
We see that every fraction is a multiple of some unit fraction:
3 5 |
= 3 × |
1 5 |
= |
1 5 |
+ |
1 5 |
+ |
1 5 |
. |
2 eighths + 3 eighths are 5 eighths. The unit is |
1 8 |
.
|
This illustrates the following principle:
In addition and subtraction, the units must be the same.
We will see this in Lesson 24. In any fraction, the denominator names the unit.
Example 4. 1 is how many fifths?
Answer. |
5 5 |
. |
|
|
|
1 5 |
is contained in 1 five times. |
Similarly,
And so on. We may write 1 with any denominator. Which is to say, we may decompose 1 into any parts: Halves, thirds, fourths, fifths, millionths.
Example 5. Add, and express the sum as an improper fraction: |
5 9 |
+ 1. |
Answer. |
5 9 |
+ 1 = |
5 9 |
+ |
9 9 |
= |
14 9 |
. |
|
Answer. Since 1 = |
8 8 |
, then |
5 8 |
+ |
3 8 |
= |
1. |
Equivalently, since finding what number to add is subtraction,
3 8 |
is called the complement of |
5 8 |
. |
3 8 |
completes |
5 8 |
to make 1. |
Example 12. How much is |
1 − |
1 3 |
? |
1 3 |
plus |
2 3 |
= |
3 3 |
, which is 1. |
Example 13. 1 − |
2 5 |
= |
3 5 |
. |
When we add |
3 5 |
to |
2 5 |
, we get 1. |
Example 14. How much
is 6 |
3 4 |
− |
1 4 |
? |
Answer. 6 |
2 4 |
. The 6 is not affected. |
Example 15. How much is 6 |
4 4 |
− |
1 4 |
? |
Look at the fact:
We are subtracting |
1 4 |
-- which is less than 1 -- from 7. The answer |
therefore falls beween 6 and 7. And |
3 4 |
is the complement of |
1 4 |
. |
Compare
In other words:
Whenever we subtract a proper fraction from a whole number greater than 1, the answer will be a mixed number which is one whole number less, and whose fraction is the complement of the proper fraction.
Example 16.
| 5 − |
1 3 |
= 4 |
2 3 |
4 is one less than 5. And |
2 3 |
is the complement of |
1 3 |
. |
On the other hand, we could say that we can only subtract thirds from thirds. Therefore we must create thirds by breaking off 1 from 5
5 − |
1 3 |
= 4 |
3 3 |
− |
1 3 |
= 4 |
2 3 |
. |
Example 17. 9 − |
2 5 |
= 8 |
3 5 |
. |
We could check this by adding:
At this point, please "turn" the page and do some Problems.
or
Continue on to the next Section.
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